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Abstract

A genus in algebraic topology is a ring homomorphism ϕ from some cobordism ring MG ∗ of manifolds to a commutative ring R. The most famous example is probably the signature of an oriented manifold, but the Atiyah-Bott-Shapiro genus ϕ: MSpin ∗ → KO ∗ [M. F. Atiyah, R. Bott and A. Shapiro, Topology 3 (1964), suppl. 1, 3–38; MR0167985 (29 #5250)] is more important in algebraic topology. Here MSpin ∗ is the Spin cobordism ring, and KO ∗ is the coefficient ring for real K-theory. One of the great achievements of 20th-century mathematics, the Atiyah-Singer index theorem, says that ϕ(M) can be calculated analytically, by taking the index of the Dirac operator on a manifold M. In the mid 1980’s, S. Ochanine [Topology 26 (1987), no. 2, 143–151; MR0895567 (88e:57031)] introduced his elliptic genus, which assigns a level 2 modular form to an oriented manifold M. This was generalized by E. Witten [in Elliptic curves and modular forms in algebraic topology (Princeton, NJ, 1986), 161–181, Lecture Notes in Math., 1326, Springer, Berlin, 1988; see MR0970278 (91a:57021)], who constructed a new elliptic genus by pretending that the Atiyah-Singer index theorem and related results could be applied to calculate the index of the S 1-equivariant Dirac operator on the infinite-dimensional manifold ΛM, the free loop space of a compact manifold M